# Astronomy 310/510: Lecture 2: In stars, hydrostatic equilbrium means pressure out cancels gravity in.

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1 Astronomy 310/510: Lecture 2: Newton s laws, his second law in particular: F = ma. If a = 0, then no net forces act. In stars, hydrostatic equilbrium means pressure out cancels gravity in. When pressure exceeds gravity, layer accelerates out. When pressure is less than gravity, layer falls in. Layer can be a spherical shell or an individual mass element. But accelerate in which direction? This direction is described most generally by spherical harmonics: a radial, polar and azimuthal component. Radial δr and a component that depends on polar 0 < θ < π and azimuthal 0 < φ < 2π angle. δr = δr(r)y (θ, φ). These spherical harmonics described by 3 numbers, n, l, m. l = m = 0: radial mode: Classical Cepheid pulsation. l = 0, 1, 2,... and m = l + 1, l + 2,..., l 1. Nodes: fundamentla, first and second overtones. n = number of nodes radially outward from the Sun s center. m number of nodes found around the equator. l number of nodes found around the azimuth (ie from north to south). Sound waves travel due to variations in pressure: pressure is the restoring force. When pressure is the restoring force for these waves in stars, the waves are called p-mode waves. For example, the p 2 mode with l = 4, m = 3 has two nodes from the center to the surface, 4 nodes going from north to south and 3 nodes around the equator. For pressure modes, a crucial parameter is the sound speed: more frequent collisions: faster, sound speed. Higher temperature, faster sound speed, higher density, faster sound speed, lighter gases, faster sound speed. 1

2 Oscillation pattern at the surface propogates in a continuous way toward the stellar center. Study of the surface pattern allows the characterization of the oscillation throughout the star. Radial pulsations: standing waves in the interior of the star. Non-radial pulsations: sound waves propogate horizontally as well as radially to produce waves that travel around the star. Sound waves travelling from the center are reflected from the surface of the star and then around the star. The more often a wave returns to the surface, the less deeply it pnenetrates before being turned back. Waves reflected only a few times from the surface probe much deeper depths into the Sun. Different oscillations penetrate to different depths and hence are governed by the sound speed in those layers. Many different modes can be excited. A mixture of these is what we observe on the surface of the Sun: the surface layer of the Sun is moving back and forward which can be observed by Doppler shifts or actually measuring the light variations. Any function can be expressed as a sum of sines and cosines of different frequencies. Decompose the actual motion of the Sun into its consitutuent modes: use Fourier analysis to decompose the total signal into its constituent notes or modes. A Fourier transform converts a signal into its constituent modes. Compare the oscillations predicted from theoretical models with those observed as above to learn about the structure of the Sun s interior. Need precise measurements of the frequency of each mode of oscillation. Excited modes can grow and decay as well: some modes have a lifetime of a day or two whilst others decay over a period of months. Observe the Sun in line-of-sight Doppler velocity measurements and/or in measurements of variations of its continuum intensity of radiation by compression of the radiating gas by the solar oscillations. Wave motions with the largest horizontal scales are detectable in which light from the whole disk is collected and analyzed as a single time series. 2

3 Motions on a smaller horizontal scale require spatially resolved measurements by separately observing different parts of the disk. Need long uninterrupted time series for precise estimation of frequencies. GONG: network of small telescopes making 1024x1024 pixel resolved observations. SOHO: from space: MDI, VIRGO, GOLF in decreasing order of resolution. In Doppler velocity, amplitudes of the individual modes are 10cms 1 or smaller with a total signal of about 1kms 1. Largest amplitude modes have frequencies of the order of 3mHz. BiSON: integrated light of the Sun observations. Observe the Sun at a particular wavelength or at a particular line eg. the Fraunhoffer line or H α. What we observe is the integrated effect of the oscillation at all points on the surface, that is a superposition of all modes excited in the Sun. This is a periodic motion and can, by the Fourier theorem, be decomposed into its constituent modes, each with a unique oscillation frequency. In BiSON, instruments are sensitive to the blue and red parts of the Fraunhoffer line, giving I R, I B as a function of time. Construct R = (I B I R )/(I B + I R ). Multiply this by a factor to put it in units of velocity. This is an integrated approach. Can also image the sun and produce a dopplergram of the Sun s image: that is the velocity of the surface as a function of surface position as a function of time. Typical solar oscillation mode has a very low amplitude variation δl/l of only 10 6 and surface velocities of about 10cms 1. Oscillations observed in the Sun fall into two categories: Modes with periods ranging from 3-8 minutes and very short horizontal wavelengths (0 < l < 1000): these are the solar five minute oscillations and are p-modes. 3

4 Modes with longer periods of about 160 minutes - could be g modes with l 1 4. Still controversial. Five minute oscillations concentrated below the photosphere within the Sun s convection zone. g modes found deep in the solar interior. The horizontal wavelength of a p mode is given roughly by λ h = 2πr l(l + 1), where r is the radial distance from the star. The acoustic frequency at this depth is S l = 2π timeforsoundtotravelλ h, S l = γp ρ l(l + 1). r Thus the acoustic frequency of a mode depends on the structure of the star/sun. By the ideal gas law, the speed of sound is proportional to the square root of the temperature and decreases with increasing r. The frequency of a p mode is determined by the average value of S l in the regions of the star where the oscillations are most energetic. In non-rotating stars, the pulsation period and frequency depends only on the number of radial nodes and on the integer l. It does not depend on m which has no physical significance in the absence of rotation. Rotation introduces axes of rotation and an equator. With rotation, pulsation frequencies for modes with different values of m are split as the travelling waves move with or against the rotation. The amount of splitting depends on the speed of rotation or the angular rotation frequence Ω of the star. Frequency splitting provides a powerful way to measure the rotation rate of the Sun s interior. Modes of low degree (l = 0 3) are observable with integrated light observations and are sensitive to conditions throughout the entire Sun, including the energy generating core. Spatially resolved observations have yielded information about p modes to degree several thousand but these modes only give information on conditions near the surface. g modes have not been unambiguously detected. From Doppler or intensity fluctuations and Fourier transformations, get frequencies. Compare to models or invert. 4

5 Get the sound speed as a function of position in the Sun. Sound speed changes abruplty at the border between the convection and radiation zones: helioseismology has determined the location of the base of the convection zone. Below this sound speed is determined by opacity: resistance to radiation flow: thus inversion of observed frequencies which yield seismically determined frequencies permit errors in the theoretically determined opacity to be obtained. Opacity is one of the main atomic physics inputs into the modeling of all types of stars. Dip int he soun dspeed at the center of the Sun is a signature of the fusion of hydrogen to helium (change in mean molecular weight). Studying the sound speed in the partial ionization regions in the outer Sun can be used to determine the fractional helium abundance in the Solar convection zone. Different (lower) to that inferred from Big Bang Theory: gravitational settling of helium etc. Detailed information about rotation as a function of depth in the Sun. 5

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